MathDB

2

Part of 2013 International Zhautykov Olympiad

Problems(2)

All n which divide one of the two numbers

Source: International Zhautykov Olympiad 2013 - D1 - P2

1/17/2013
Find all odd positive integers n>1n>1 such that there is a permutation a1,a2,a3,…,ana_1, a_2, a_3, \ldots, a_n of the numbers 1,2,3,…,n1, 2,3, \ldots, n where nn divides one of the numbers ak2−ak+1−1a_k^2 - a_{k+1} - 1 and ak2−ak+1+1a_k^2 - a_{k+1} + 1 for each kk, 1≤k≤n1 \leq k \leq n (we assume an+1=a1a_{n+1}=a_1).
modular arithmeticnumber theory proposednumber theory
The sum of lengths does not exceed the perimeter

Source: International Zhautykov Olympiad 2013 - D2 - P2

1/17/2013
Given convex hexagon ABCDEFABCDEF with AB∥DEAB \parallel DE, BC∥EFBC \parallel EF, and CD∥FACD \parallel FA . The distance between the lines ABAB and DEDE is equal to the distance between the lines BCBC and EFEF and to the distance between the lines CDCD and FAFA. Prove that the sum AD+BE+CFAD+BE+CF does not exceed the perimeter of hexagon ABCDEFABCDEF.
geometryhexagon